The Circle Inversion Theorem

The Circle Inversion Theorem states that in a circle inversion—a transformation where points are mapped based on their distance from a fixed center—lines and circles follow predictable patterns. Specifically, a line not passing through the center transforms into a circle that passes through the center, while a circle passing through the center maps into a straight line. Conversely, circles not passing through the center remain circles, but their size and position change according to the inversion. This theorem helps understand how shapes relate under inversion, revealing deep geometric symmetries and simplifying complex configurations.